A jar contains 3 pink balls, 6 blue balls, and 3 red balls.
Event A = drawing a red ball on the first draw
Event B = drawing a pink ball on the second draw
If two balls are drawn from the jar, one after the other without replacement, what is P(A and B) expressed in simplest form
To find the probability of both event A (drawing a red ball on the first draw) and event B (drawing a pink ball on the second draw), we calculate:
P(A and B) = P(A) * P(B|A)
P(A) = (number of red balls) / (total number of balls)
Since there are 3 red balls and a total of 3 pink balls, 6 blue balls, and 3 red balls, the total number of balls is 3 + 6 + 3 = 12.
P(A) = 3 / 12 = 1 / 4
Now, we need to find P(B|A), which means the probability of drawing a pink ball on the second draw given that we have already drawn a red ball on the first draw.
After drawing a red ball on the first draw, we will have 2 red balls left in the jar and a total of 11 balls remaining.
P(B|A) = (number of pink balls remaining) / (total number of balls remaining)
Since we have already drawn one red ball, we have 2 pink balls and 11 total balls remaining.
P(B|A) = 2 / 11
Now we can calculate the probability of both events A and B occurring:
P(A and B) = P(A) * P(B|A) = (1/4) * (2/11)
To express the probability in simplest form, we divide both the numerator and denominator by their greatest common divisor (GCD):
P(A and B) = (1/4) * (2/11) = (1 * 2) / (4 * 11) = 2/44
Since 2/44 can be simplified by dividing both the numerator and denominator by 2, we get:
P(A and B) = 1/22
Therefore, the probability of drawing a red ball on the first draw and a pink ball on the second draw is 1/22.
To calculate the probability of Event A and Event B occurring, we need to find the probability of drawing a red ball on the first draw and a pink ball on the second draw.
Let's find the probability of Event A first.
There are a total of 12 balls in the jar, and of those, 3 are red. So, the probability of drawing a red ball on the first draw is 3/12.
Now, after the first ball is drawn, there are 11 balls left in the jar, and of those, 3 are pink. So, the probability of drawing a pink ball on the second draw is 3/11.
To find the probability of both events occurring, we multiply the probabilities together:
P(A and B) = (3/12) * (3/11)
To simplify this, we can cancel out a common factor of 3:
P(A and B) = (1/4) * (1/11)
Therefore, the probability of both Event A and Event B occurring is 1/44.
3/12 = ?
Since on red ball is already gone,
3/11 = ?
The probability of both/all events occurring is determined by multiplying the probabilities of the individual events.